## Critical droplets and metastability for a Glauber dynamics at very low temperatures.(English)Zbl 0722.60107

Summary: We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two- dimensional torus under a positive magnetic field in the limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins $$+1$$ in a sea of spins -1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down ($$-\underline 1$$) the pattern of evolution leading to the more stable configuration with all spins up $$(+\underline 1)$$ approaches, as the temperature vanishes, a metastable behavior: the system stays close to $$-\underline 1$$ for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to $$+\underline 1$$ in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 82C27 Dynamic critical phenomena in statistical mechanics

### Keywords:

ferromagnetic spin system; Glauber dynamics; metastability
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### References:

 [1] [Bra] Brassesco, S.: Tunelling for a non-linear heat equation with noise. Preprint (1989) [2] [CGOV] Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: A pathwise approach. J. Stat. Phys.35, 603–634 (1984) · Zbl 0591.60080 [3] [COP] Cassandro, M. Olivieri, E., Picco, P.: Small random perturbations of infinite dimensional dynamical systems and nucleation theory. Ann. Inst. Henri Poincaré (Phys. Theor.)44, 343–396 (1986) · Zbl 0598.35133 [4] [GOV] Galves, A., Olivieri, E., Vares, M. E.: Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab.15, 1288–1305 (1987) · Zbl 0709.60058 [5] [EGJL] Eston, V. R., Galves, A., Jacobi, C. M., Langevin, R.: Dominance switch between two interacting species and metastability. Preprint (1988) [6] [HMM] Huiser, A. M. J., Marchand, J.-P., Martin Ph.A.: Droplet dynamics in two-dimensional kinetic Ising model. Helv. Phys. Acta,55, 259–277 (1982) [7] [Kes] Kesten, H.: Percolation Theory for Mathematicicians. Boston-Basel: Birkhauser 1982 [8] [KN] Kipnis, C., Newman, C. M.: The metastable behavior of infrequently observed, weakly random, one dimensional diffusion processes. SIAM J. Appl. Math.45, 972–982 (1985) · Zbl 0592.60063 [9] [Lig] Liggett, T. M.: Interacting particle systems. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0559.60078 [10] [LS] Lebowitz, J. L. Schonmann, R. H.: On the asymptotics of occurrence times of rare events for stochastic spin systems. J. Stat. Phys.48, 727–751 (1987) · Zbl 1084.82521 [11] [NCK] Newman, C. M., Cohen, J. E., Kipnis, C.: Neo-darwinian evolution implies punctuated equilibria. Nature315, 400–401 (1985) [12] [MOS] Martinelli, F., Olivieri, E., Scoppola, E.: Small random perturbation of finite and infinite dimensional systems: Unpredicted of exist times. Preprint (1988) [13] [PL] Penrose, O. Lebowitz, J. L.: Molecular theory of metastability: An update. Appendix to the reprinted edition of the article ”Towards a rigorous molecular theory of metastability” by the same authors. In: Fluctuation Phenomena (second edition). Montroll, E. W., Lebowitz, J. L. (eds.) Amsterdam: North-Holland Physics Publishing 1987 [14] [Sch] Schonmann, R. H.: Metastability for the contact process. J. Stat. Phys.41, 445–464 (1985) · Zbl 0646.60108
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